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computational geometry introduction
Mark de Berg
TU/e
IPA Basic Course—Algorithms
introduction• what is CG• applications
randomized incremental construction• easy example: sorting• a general framework• applications to geometric problems
and more …
TU/e
Computational Geometry
Computational Geometry:
Area within algorithms research dealing with spatial data (points, lines, polygons, circles, spheres, curves, … )
design of efficient algorithms and data structures for spatial data
elementary operations are assumed to take O(1) time and be available
focus on asymptotic running times of algorithms
computing intersection point of two lines, distance between two points, etc
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Map overlay
land usage annual rainfall
overlay
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Computational Geometry
example: compute all intersections in a set of line segments
naïve algorithm
for i = 1 to n
for j =i+1 to n
do if si intersects sj
then report intersection
Running time: O(n2)
Can we do better ?
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Computational geometry: applications (1)
geographic information systems (GIS)
Mississippi delta
MemphisMemphis
MemphisMemphis
ArlingtonArlington
30 cities:
430 = 1018 possibilities
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Computational geometry: applications (2)
computer-aided design and manufacturing (CAD/CAM)
3D model of power plant (12.748.510 triangles)
motion planning
virtual walkthroughs
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Computational geometry: applications (3)
3D copier
surface reconstruction
3D scanner
3D printer
1
2
34 5 6 7 8
9
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Computational geometry: applications (4)
Other applications of surface reconstruction
digitizing cultural heritage
custom-fit products
reverse engineering
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Computational geometry: applications (5)
geographic information systems
computer-aided design and manufacturing
robotics and motion planning
computer graphics and virtual reality
databases
structural computational biology
and more …
30 45 age
salary
30.000
50.000
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Computational Geometry: algorithmic paradigms
Deterministic algorithms plane sweep geometric divide-and-conquer
Randomized algorithms randomized incremental construction random sampling
today’s focus
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EXERCISES
1. Let S be a set of n line segments in the plane. a) Design an algorithm that decides in O (n log n) time if
there are any intersections in S. Hint: Sort the endpoints on y-coordinates, and handle
them in that order in a suitable manner.b) Extend the algorithm so that it reports all intersections in
time O ((n+k) log n), where k is the number of intersections.
For the experts:
Let u (n ) be the maximum number of pairs of points at distance 1 in a set of n points in the plane. Prove that u(n) = Ω (n log n).
Let D be a set of n discs in the plane. Prove that the union complexity of D is O (n ).
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Plane-sweep algorithm for line-segment intersection
1
23
45 6
sweep lineinsert segment 1insert segment 2insert segment 3
insert segment 4insert segment 5delete segment 1
When two segments intersect, then there is a horizontal line L such that
the two segments both intersect L
they are neighbors along L
Ordering along L: 1,23,1,23,1,4,23,5,1,4,21 3,5,4,2
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Plane sweep
SegmentIntersect
1. Sort the endpoints by y-coordinate. Let p1,…,p2n be the sorted set of endpoints.
2. Initialize empty search tree T.
3. for i = 1 to 2n
4. do { if pi is the left endpoint of a segment s
5. then Insert s into the tree T
6. else Delete s from the tree T
7. Check all pairs of segments that become
neighbors along the sweep line for intersection,
report if there is an intersection. }
Running time ?
Extension to reporting all intersections ?
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COFFEE
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Lecture II: Randomized incremental construction
ParanoidMax (A)(* A [1..n ] is an array of n numbers *)1. Randomly permute the elements in A2. max := A[1]3. for i := 2 to n4. do if A [i ] > max5. then { max := A[i ]6. for j = 1 to i -1 7. do if A[i ] > max then error }8. return max
Warm-up exercise:
Analyze the worst-case and expected running time of the following algorithm.
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Worst-case running time
Worst-case = O(1) + time for lines 2—7
= O(1) + ∑ 2≤i≤n (time for i-th iteration)
= O(1) + ∑ 2≤i≤n ∑ 1≤j≤i-1 O(1)
= O(1) + ∑ 2≤i≤n O(i)
= O(n2)
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Expected running time
E [ looptijd ] = O(1) + E [ time for lines 2—7 ]
= O(1) + E [ ∑ 2≤i≤n (time for i-th iteration) ]
= O(1) + ∑ 2≤i≤n E [ time for i-th iteration ]
= O(1) + ∑ 2≤i≤n { Pr [ A[i] ≤ max { A[1..i-1 ] } ] ∙ O(1)
+ Pr [ A[i] > max { A[1..i-1 ] } ] ∙ O(i) }
= O(1) + ∑ 2≤i≤n { 1 ∙ O(1) + (1/i ) ∙ O(i) }
= O(n )
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EXERCISE
1. Give an algorithm that puts the elements of a given array A[1..n] into random order. Your algorithm should be such that every possible permutation is equally likely to be the result.
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Sorting using Incremental Construction
Input: numbers x1,…,xn
Output: partitioning of x-axis into a collection T of empty intervals defined by the input points
x3 x1 x5 x4 x2
A geometric view of sorting
Incremental Construction: “Add” input points one by one,
and maintain partitioning into intervals
x3 x1 x5 x4 x2
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Sorting using Incremental Construction
x3 x1 x5 x4 x2
IC-Sort (S)
1. T := { [ -infty : +infty] }
2. for i := 1 to n
3. do { Find interval J in T that contains xi and remove it.
4. Determine new intervals and insert them into T. }
for each point, maintain pointer to interval that contains it
for each interval in T, maintain conflict list of all points inside
list is empty
list: x4, x5
How?
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Sorting using Incremental Construction
IC-Sort (S)
1. T := { [ -infty : +infty] }
2. for i := 1 to n
3. do { Find interval J in T that contains xi and remove it.
4. Determine new empty intervals and insert them into T.
5. Split conflict list of J to get the two new conflict lists. }
for each point, maintain pointer to interval that contains it
for each interval in T, maintain conflict list of all points inside
TOTAL TIME: O ( total size of conflict lists of all intervals that ever appear)
WORST-CASE:
Time analysis:
O(n2)
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Sorting using Incremental Construction
RIC-Sort (S)
1. Randomly permute input points
2. T := { [ -infty : +infty] }
3. for i := 1 to n
4. do { Find interval J in T that contains xi and remove it.
5. Determine new empty intervals and insert them into T.
6. Split conflict list of J to get the two new conflict lists. }
for each point, maintain pointer to interval that contains it
for each interval in T, maintain conflict list of all points inside
TOTAL TIME: O ( total size of conflict lists of all intervals that are created)
EXPECTED TIME: ??
Time analysis:
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An abstract framework for RIC
S = set of n (geometric) objects; the input
C = set of configurations
D: assigns a defining set D(Δ) S to every configuration Δ є C
K: assigns a killing set K(Δ) S to every configuration Δ є C
4-tuple (S, C, D, K) is called a configuration space
configuration Δ є C is active over a subset S’ S if:
D(Δ) S’ and K(Δ) S’ = o
T(S) = active configurations over S; the output
∩∩
∩
∩ ∩ /
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Sorting using Incremental Construction
RIC(S)
1. Compute random permutation x1,…xn of input elements
2. T := { active configurations over empty set }
3. for i := 1 to n
4. do { Remove all configurations Δ with xi є K(Δ) from T.
5. Determine new active configurations; insert them into T.
6. Construct conflict lists of new active configurations. }
for each xi, maintain pointers to all configurations Δ such that xi є K(Δ)
for each configuration Δ in T, maintain conflict list of all xi є K(Δ) Theorem: Expected total size of conflict lists of all created configs:
O( ∑ 1≤i≤n (n / i2) ∙ E [ # active configurations in step i ] )
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EXERCISE
1. Use the theorem to prove that the running time of the RIC algorithm for sorting runs in O( n log n ) time.
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LUNCH
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Lecture III
RIC: more applications
Voronoi diagrams and Delaunay triangulations
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Principia Philosophiae (R. Descartes, 1644).
Voronoi diagram
The universe according to Descartes
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Construction of 3D height models (1)
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height?
Construction of 3D height models (2)
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Construction of 3D height models (3)
Better: use interpolation to determine the height
triangulation
983
983
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Construction of 3D height models (4)
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long and thin triangles are bad
try to avoid small angles
What is a good triangulation?
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Construction of 3D height models (5)
Voronoi diagram
Delaunay triangulation:
triangulation that maximizes the minimum angle !
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Voronoi diagrams and Delaunay triangulations
P : n points (sites) in the plane
V(pi ) = Voronoi cell of site pi
= { q є R2 : dist(q,pi ) < dist(q,pj ) for all j ≠i }
Vor(P ) = Voronoi diagram of P = subdivision induced by Voronoi cells
DT(P) = Delaunay triangulation of P = dual graph of Vor(P)
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EXERCISES
3. Prove: a triangle pi pj pk is a triangle in DT(P) if and only if its circumcircle C (pi pj pk ) does not contain any other site
4. Prove: number of triangles in DT(P) is at most 2n-5
Hint: Euler’s formula for connected planar graphs: #vertices — #edges + #faces = 2
3. Design and analyze a randomized incremental algorithm to construct DT(P).
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COFFEE
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RIC – Limitations of the framework
Robot motion planning
What is the region that is reachable for the robot?
Lecture IV: wrap-up
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RIC – Limitations of the framework (cont’d)
The single-cell problem:
Compute cell containing the origin
Idea: compute vertical decomposition for the cell using RIC ?!
TU/e Spatial data structures store geometric data in 2-, 3- or higher dimensional space such that certain queries can be answered efficiently
applications: GIS, graphics and virtual reality, …
Examples:
range searching
nearest-neighbor searching
point location
more computational geometry
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plane sweep
parametric search
arrangements
geometric duality, inversion, Plücker coordinates, …
kinetic data structures
core sets
Davenport-Schinzel sequences
…
and more … and more …
